It appears there has not been much innovation in depilatory technology, since this product follows the same basic formulation that has been used for years. The potassium thioglycolate in combination with calcium hydroxide react to break down the disulfide bonds in hair, allowing for easy removal with wiping or rinsing.
The pH of a depilatory is typically around 12.5. Not many ingredients can remain stable in the extreme alkaline environment found in depilatories, which is why ceteareth-20 (and) cetearyl alcohol is the most common emulsifier/stabilizer combination used in such formulations.
Urea is used as a humectant for moisturization in most skin care products, but in a depilatory its primary function is to swell the hair to speed up depilation. Mineral oil and lanolin coat the skin and help to prevent irritation.